fundamentals of musical acoustics · fundamentals of musical acoustics! what is sound?! air...
TRANSCRIPT
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Fundamentals of Musical Acoustics
What is sound?
air pressure
time
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Time
something else (air pressure)
time ?
t = 0
These variations in air pressure over time can be decomposed into
sine waves with
amplitude and frequency
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Peak KE
Peak PE
Simple Harmonic Motion (Makes sine waves)
t
y
x θ cos θ
sin θ 1
Sinusoids y = sin θ x = cos θ
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1
-1
π
2π
3π
4π
a
t
a = sin(t)
a = cos(t)
A
A
-A
Amplitude
a = A sin(t)
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y
x θ A cos θ
A sin θ A
y = A sin θ x = A cos θ
Frequency
How frequently does the sin/cos complete a whole cycle?
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C B
A
D
F H G
E
π 2π 3π t
π/2 3π/2 5π/2 phase φ
180 360/0 180 90 270 90
radians
degrees
a t
A C G B D E F A C G B D E F H H A
Frequency
1 cycle 2π
1 cycle / sec. 2π / sec.
f cycles / sec. . 2π / sec. f
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Radian Frequency
ω = 2πf
f = ω/2π
y
x
θ A
y = A cos (2πft) x = A cos (2πft)
θ = 2πft
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a
t
a = A sin(2πft)
a = A cos(2πft)
a
t
Period T
T = period
f = 1 T
fraction of a second in which one cycle repeats
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Phase
φ
a = A sin(2πft + φ)
a = A sin(2πft + φ)
f = 1 T
φ
2π = 1 period T
t
A
-A
phase
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Two Domains of Description
Time
Frequency
a
t
a
f
(Amplitude) A
a
f
freq (Phase) φ
π
-π a = A sin (2πf + φ)
Frequency Domain Representation
freq
f
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Frequency Domain What can I hear?
Fletcher & Munson
Frequency (Hz)
IL (dB)
a
f F 2F 3F 4F 5F
Modes of Vibration
N = node
N
N N
N N N
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Joseph Fourier (1768 - 1830)
f(t) = A sin(2πnFt + φ ) n n Σ n=0
∞
periodic function f(t) = f(t+T)
T period
Integral multiple harmonics A1
A2 A3
A4 A5
A6 A7
A8 A9 A10 A12
F 2F 3F 4F 5F 6F 7F 8F 9F 10F 11F 12F A11
Joseph Fourier (1768 - 1830) Harmonic Series
Modes of Vibration
Fundamental
Summation
F 2F 3F 4F 5F 6F 7F 8F 9F 10F 11F 12F
2F
3F 4F 5F 6F
F
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Superposition of Harmonics
Ongoing Superposition / Interference
constructive
destructive 180° out of phase
Identical Frequencies Different Frequencies
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a
f F 2F 3F 4F 5F 6F
1 2 3 4 5 6 1 2 3 4 5
harmonic overtone
harmonic
a
inharmonic partials
f
a
f f0
f f0 behind φ
π
-π
Resonance
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Resonance
Frequency (Hz)
Time
Amplitude
Amplitude
Input: Amplitude
Output:
a
f f0
.7071 1.0
Δf Δf f0 Q =
a
f f0
High Q
Low Q
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Frequency (kHz)
Input admittance level (10 dB/div)
FIG. 10.14 Input admittance (driving point mobility) of a Guarneri violin driven on the bass bar side (Alonso Moral and Jansson, 1982b)
Time Domain
Real world Sounds
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a
t
a
f
formants
cutoff
attack sustain decay
envelopes
temporal envelope
spectral envelope
Ongoing Spectral Change Pressure Wave
Am
plitu
de
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3-D graphs a
f
t
low
high
soft
loud
start stop
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Time
a
t
a
f Frequency